Stable rationality of higher dimensional conic bundles

Abstract

We prove that a very general nonsingular conic bundle $X\to {\mathbb{P}}^{n-1}$ embedded in a projective vector bundle of rank $3$ over ${\mathbb{P}}^{n-1}$ is not stably rational if the anti-canonical divisor of $X$ is not ample and $n\ge 3$.

Final version. To appear in Epijournal de Geometrie Algebrique

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